The focus of Project 1 is the development of stochastic models of carcinogenesis and experimental data using these models. The specific aims of this project fall into three groups. 1. The hazard function and analyses of time to tumor data. In this part of the project, we will investigate the properties of the hazard function of the two-mutation clonal expansion model and extensions of it. In particular, we propose to investigate issues of identifiability, goodness- of-fit, incorporation of time to tumor-detection, patterns of exposure to environmental agents, approximations to the hazard function, and covariate measurement errors. 2. Number and size distribution of intermediate (premalignant) lesions and analyses of initiation-promotion experiments. In this part of the project, we will develop models for the analyses of altered foci in the rodent liver or papillomas on the mouse skin. In particular, we will investigate shape-free methods for stereological reconstruction and analyses of foci in the rodent liver and methods for analyses of correlated longitudinal data on papillomas on the mouse skin. 3. Development of efficient computational algorithms. We intend to develop efficient computer algorithms and software, including highly computer-intensive methods such as Markov Chain Monte Carlo methods, for analyzing data using the models developed in 1 and 2.